The combinatorics behind the leading Kazhdan-Lusztig coefficients of braid matroids
Abstract
Ferroni and Larson gave a combinatorial interpretation of the braid Kazhdan-Lusztig polynomials in terms of series-parallel matroids. As a consequence, they confirmed an explicit formula for the leading Kazhdan-Lusztig coefficients of braid matroids with odd rank, as conjectured by Elias, Proudfoot, and Wakefield. Based on Ferroni and Larson's work, we further explore the combinatorics behind the leading Kazhdan-Lusztig coefficients of braid matroids. The main results of this paper include an explicit formula for the leading Kazhdan-Lusztig coefficients of braid matroids with even rank, a simple expression for the number of simple series-parallel matroids of rank k + 1 on 2k elements, and explicit formulas for the leading coefficients of inverse Kazhdan-Lusztig polynomials of braid matroids. The binomial identity for the Abel polynomials plays an important role in the proofs of these formulas.
Keywords
Cite
@article{arxiv.2311.06929,
title = {The combinatorics behind the leading Kazhdan-Lusztig coefficients of braid matroids},
author = {Alice L. L. Gao and Nicholas Proudfoot and Arthur L. B. Yang and Zhong-Xue Zhang},
journal= {arXiv preprint arXiv:2311.06929},
year = {2023}
}